# Different definitions of the rank of a module

I have seen different definitions of a rank of a module $$M$$ over a commutative ring $$R$$.

1. Here in nLab, for quite general modules, the rank is defined locally at $$p\in \mathrm{Spec}(R)$$ as the dimension over the residue field $$\kappa(p)$$ of the $$\kappa(p)$$ vector space $$M_p/pM_p$$. When $$M$$ is of finite type, this is the local minimum of generators of $$M_p$$ over $$R_p$$.

2. The usual definition of the rank of a module when this module is projective of finite type is locally the rank of the free module $$M_p$$ over $$R_p$$. (see mathoverflow.net/questions/29993 for instance). This function rank is locally constant, and therefore when $$R$$ is a domain, is the dimension of the vector space $$M\otimes K$$ over $$K=R_0$$.

3. When $$R$$ is a domain, Matsumura defines the rank of a quite general module $$M$$ by the maximum number of independent elements, that is the dimension of the vector space $$M\otimes K$$ over $$K=R_0$$.

Clearly, all these definitions coincides when $$M$$ is projective of finite type. When $$R$$ is a domain, and $$M$$ only of finite type, I see no reason why they should outside of the ideal $$0$$. I mean, in that case, to be projective for $$M$$ is the same as to be locally free. But if $$M$$ is not projective, the local minimum of generators of $$M_p$$ (definition 1) does not necessarily coincides with the global maximum of independent elements (definition 3), right?

Did I misunderstand something ? If I didn't, which definition should be used? And in the case of $$M$$ is only of finite type?

EDIT: In case the ring $$R$$ is noetherian and $$M$$ is of finite type, we have the following generalization of Matsumura's definition, if $$Q$$ is the total ring of fraction of $$R$$, with the equivalence of all the following statements:

1. $$M$$ is of rank $$r$$.

2. $$M\otimes Q$$ is a free rank $$r$$ module over $$Q$$.

3. The maximum number of independent elements of $$M$$ is $$r$$.

4. There exists a submodule $$N$$ of $$M$$ such that $$M/N$$ is torsion.

5. For all prime $$p$$ associated to $$R$$, $$M_p$$ is free of rank $$r$$ over $$R_p$$.

I like this last definition because this is the most global one when $$R$$ is not a domain (provided the module actually have a finite rank ...), and it is additive on exact sequence when two of the terms actually have finite rank.

• All definitions are widely used. For example, when speaking of the rank of the Mordell-Weil group of an elliptic curve, one means 3). It's usually clear in context what is meant.
– abz
Jan 5, 2014 at 17:27
• Thank you for your information. Usually I don't like when under the same name different definitions give different results. But if it is not a problem in everyday work ... In any case, I was thinking that when M is locally free of finite type only, 1 & 2 are identical. When R is a domain in addition, then they all are identical. So my only problem is when R is a domain, M finite type not projective, 2. makes sense if we define the rank of Mp over Rp using 3., but then it is different from 1. IMO, 1. is not really about the rank because it cancels too much information, and more about generators Jan 5, 2014 at 19:37
• Though one uses the definitions 2 or 3 for rank in general, here is a nice result and usefulness of the first definition due to O. Forster and R. G. Swan. If $M$ is a finitely generated module over a Noetherian ring $R$, define $\mu(M)$ to be the maximum of $r_p(M)+\dim R/p$ as $p$ varies over all prime ideals. Then $M$ can be generated by $\mu(M)$ elements. Jan 5, 2014 at 21:36
• @Mohan: I didn't know that result and find it very striking. Could you please leave an answer that describes it? Jan 6, 2014 at 0:15
• @PeteL.Clark: I have already mentioned the result and you can easily find their papers on the web. What else would you like to know? There are also generalizations of these for polynomial rings, called Eisenbud-Evans conjectures, proved by Sathaye and myself (this was my Ph. d. thesis). Jan 6, 2014 at 16:45

For a finitely generated module $M$ over a commutative ring $R$, the first definition gives a rank function $r_M: \operatorname{Spec} R \rightarrow \mathbb{N}$, whereas the third definition gives $r_M((0))$.

When $M$ is projective, $r_M$ is locally constant, so when $\operatorname{Spec} R$ is connected -- so when $R$ is a domain -- $r_M$ is constant and may be identified with its value at $(0)$. I think you are asking for reassurance that for finitely generated non-projective modules over a domain, the rank function need not be constant. That's certainly true: take $\mathbb{Z}/p\mathbb{Z}$ over $\mathbb{Z}$ (or even over $\mathbb{Z}_p$): then the rank function is $1$ at $(p)$ and otherwise $0$, so the first definition really is not the same as the third.

Is this a problem? I don't think so. Asking for terminology in mathematics to be globally consistent seems to be asking for too much: even more basic and central terms like "ring" and "manifold" do not have completely consistent definitions across all the mathematical literature: rather, they overlap enough to carry a common idea. That is certainly the case here.

Similarly, asking which of 1) and 3) is "right" doesn't seem so fruitful. It is true that the first definition records more information than the third definition. But it's just terminology, and it is often useful to have a term which records exactly the information in the third definition: e.g. the "rank of an abelian group" is a very standard and useful notion, and usually often it means 3). (I am a number theorist, and in number theoretic contexts this definition of rank is quite standard. Apparently it is less so elsewhere...) For a finitely generated module over a PID, of course the rank in the sense of 3) is exactly what you need in addition to the torsion subgroup in order to reconstruct the module. In this context the rank function 1) gives some information about the torsion but not complete information -- e.g. $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/p^2\mathbb{Z}$ have the same rank function -- so the rank function as in 1) does not seem especially natural or useful. I am (almost) sure there are other contexts where it would be natural and useful to think in terms of the rank function as in 1).

Added: Following quid's comments I checked out Fuchs's text on infinite abelian groups, and indeed the rank of an arbitrary abelian group is defined in a way so as to take $p$-primary torsion into account. (I simply didn't know this was true.) Thus the rank defined there would be the sum over all the values of the rank function in the sense of 1).

All this seems to indicate, even more than my answer, that different notions of rank proliferate.

Added Later: Quid also suggests consideration of the quantity $\operatorname{mg}(M)$, the minimal number of generators of an $R$-module $M$: this is a cardinal invariant which is (clearly) finite if and only if $M$ is finitely generated. Let $R$ be a Dedekind domain with fraction field $K$. For a maximal ideal $\mathfrak{p}$ of $R$ and a finitely generated $R$-module $M$, let $M[\mathfrak{p}^{\infty}]$ be the submodule of $M$ consisting of elements annihilated by some power of $\mathfrak{p}$. Then $M[\mathfrak{p}^{\infty}]$ is a finitely generated torsion module over the DVR $R_{\mathfrak{p}}$ and is thus a direct sum of $\operatorname{mg}(M[\mathfrak{p}^{\infty}])$ copies of $R_{\mathfrak{p}}/\mathfrak{p}^k R_{\mathfrak{p}}$. Let us define $tr(M,\mathfrak{p})$ to be this number of copies. (When $M$ is infinitely generated I believe one should also count copies of $K_{\mathfrak{p}}/R_{\mathfrak{p}}$ in a certain sense in order to recover Fuchs's $p$-primary torsion rank in the $R = \mathbb{Z}$ case. Let me omit this for now.)

Now define

$R(M) = r_M((0)) + \sum_{\mathfrak{p} \in \operatorname{MaxSpec} R} tr(M,\mathfrak{p})$

When $R = \mathbb{Z}$ then $R(M) = \operatorname{mg}(M)$ is the "total rank" in Fuchs's sense. More generally $R(M) = \operatorname{mg}(M)$ when $R$ is a PID. However, I wanted to point out that in general the function $\operatorname{mg}$ behaves rather badly. This is discussed in $\S$ 6.5.3 of my commutative algebra notes. In particular, when $M$ is finitely generated projective, $\operatorname{mg}(M) \geq R(M) (= r_M((0))$ but can be larger. However, it is much more restricted than what I knew about before reading the comments on this question. In particular, it follows from the Forster-Swan Theorem that when $M$ is projective of rank $n$ then $\operatorname{mg}(M) \in \{n,n+1\}$. (In an earlier version of this answer I knew only that $\operatorname{mg}(M) \leq 2n-1$ and "guessed" that it could be that large over suitable Dedekind domains. Not a terrible guess, perhaps, but not the most educated one either...)

• @PeteL.Clark thank you for your answer. It is clear for the comparison between 1. And 3. But what I was trying to find is a way to make 2. well defined in more general situations : for instance when $M$ is not locally free , one can still look at the maximum number of independents elements locally, right ? The difference between 1. and 2. is then a measure of the failure of local freeness for $M$. Jan 5, 2014 at 21:46
• IMO, the final remarks on abelian groups could be a bit misleading. Namely, they seem quite at odds with the treatment of 'rank like' notions in 'Infinite Abelian Groups' by Fuchs which is AFAIK a standard reference on the subject. First, rank of an abelian group there is defined differently so the 'usually' potentially is an overstatement. Second, the rank function is defined and used, via defining p-rank, for p prime, and torsion-free rank (which would be what is called rank here); rank there would be the sum of all the p-ranks and torsion-free rank.
– user9072
Jan 5, 2014 at 23:14
• @quid: What is the definition of rank given in Fuchs? Jan 6, 2014 at 0:00
• I first replied but deleted the comment upon seeing you already had edited after your comment. Thanks for taking the remark into account. Let me just add that I think still one more notion of rank for abelian groups around namely minimal cardiniality of generating set, so that all cyclic groups have rank 1, and for fin gen case the rank corresponds to the number of cyclic groups one uses when taking as orders a chain of divisors for the torsion part.
– user9072
Jan 6, 2014 at 0:25